/*
  This code can be loaded, or copied and pasted, into Magma.
  It will load the data associated to the HMF, including
  the field, level, and Hecke and Atkin-Lehner eigenvalue data.
  At the *bottom* of the file, there is code to recreate the
  Hilbert modular form in Magma, by creating the HMF space
  and cutting out the corresponding Hecke irreducible subspace.
  From there, you can ask for more eigenvalues or modify as desired.
  It is commented out, as this computation may be lengthy.
*/

P<x> := PolynomialRing(Rationals());
g := P![7, 2, -7, -2, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {17, 17, 2*w^3 - 6*w^2 - 7*w + 8}>;

primesArray := [
[4, 2, w^3 - 3*w^2 - 3*w + 4],
[7, 7, w],
[7, 7, w^3 - 3*w^2 - 4*w + 5],
[9, 3, w^3 - 4*w^2 - w + 9],
[17, 17, 2*w^3 - 6*w^2 - 7*w + 8],
[17, 17, -w^3 + 3*w^2 + 4*w - 3],
[17, 17, -w + 2],
[23, 23, 2*w^3 - 7*w^2 - 4*w + 12],
[23, 23, -w^2 + 2*w + 3],
[31, 31, -2*w^3 + 7*w^2 + 5*w - 12],
[31, 31, -w^3 + 4*w^2 + 2*w - 8],
[41, 41, 3*w^3 - 10*w^2 - 7*w + 16],
[41, 41, 2*w^3 - 7*w^2 - 5*w + 10],
[49, 7, 2*w^3 - 6*w^2 - 6*w + 9],
[71, 71, w^2 - 2*w - 2],
[71, 71, 2*w^3 - 7*w^2 - 4*w + 13],
[73, 73, 3*w^3 - 11*w^2 - 5*w + 19],
[73, 73, -4*w^3 + 13*w^2 + 10*w - 17],
[79, 79, 3*w^3 - 9*w^2 - 10*w + 13],
[79, 79, -2*w^3 + 6*w^2 + 5*w - 8],
[97, 97, 2*w^3 - 6*w^2 - 8*w + 9],
[97, 97, 2*w - 1],
[113, 113, 2*w^3 - 6*w^2 - 7*w + 6],
[113, 113, -w^3 + 3*w^2 + 2*w - 1],
[137, 137, w^3 - 2*w^2 - 5*w + 2],
[137, 137, 2*w^3 - 7*w^2 - 3*w + 12],
[137, 137, -3*w^3 + 10*w^2 + 7*w - 17],
[137, 137, -w^3 + 2*w^2 + 6*w - 2],
[167, 167, -2*w^3 + 7*w^2 + 6*w - 10],
[167, 167, 3*w^3 - 9*w^2 - 10*w + 10],
[169, 13, -2*w^3 + 7*w^2 + 2*w - 9],
[169, 13, -2*w^3 + 5*w^2 + 10*w - 4],
[193, 193, -2*w^3 + 8*w^2 + 3*w - 13],
[193, 193, -3*w^3 + 11*w^2 + 6*w - 22],
[199, 199, 5*w^3 - 16*w^2 - 14*w + 26],
[199, 199, w^3 - 2*w^2 - 7*w + 3],
[223, 223, -3*w^3 + 8*w^2 + 12*w - 8],
[223, 223, 3*w^3 - 11*w^2 - 6*w + 17],
[223, 223, 2*w^3 - 8*w^2 - 3*w + 18],
[223, 223, -4*w^3 + 13*w^2 + 9*w - 18],
[233, 233, -3*w^3 + 10*w^2 + 6*w - 16],
[233, 233, -2*w^3 + 5*w^2 + 9*w - 6],
[241, 241, -3*w^3 + 9*w^2 + 8*w - 10],
[241, 241, 4*w^3 - 12*w^2 - 13*w + 15],
[257, 257, 2*w^3 - 8*w^2 - w + 16],
[257, 257, 4*w^3 - 13*w^2 - 11*w + 18],
[257, 257, -w^3 + 5*w^2 - 2*w - 9],
[257, 257, -w^3 + 2*w^2 + 4*w + 2],
[263, 263, 4*w^3 - 14*w^2 - 9*w + 27],
[263, 263, 5*w^3 - 15*w^2 - 17*w + 25],
[263, 263, -2*w^3 + 7*w^2 + 6*w - 16],
[263, 263, -2*w^3 + 7*w^2 + 6*w - 9],
[271, 271, -w - 4],
[271, 271, -w^3 + 3*w^2 + 4*w - 9],
[271, 271, 5*w^3 - 16*w^2 - 14*w + 24],
[271, 271, -4*w^3 + 14*w^2 + 9*w - 24],
[281, 281, -2*w^3 + 6*w^2 + 5*w - 4],
[281, 281, -3*w^3 + 9*w^2 + 10*w - 9],
[281, 281, 3*w^3 - 10*w^2 - 8*w + 12],
[281, 281, w^2 - w - 8],
[311, 311, w^2 - 5],
[311, 311, -4*w^3 + 13*w^2 + 12*w - 20],
[313, 313, 6*w^3 - 19*w^2 - 18*w + 31],
[313, 313, -2*w^3 + 5*w^2 + 6*w - 6],
[337, 337, 3*w^3 - 11*w^2 - 3*w + 15],
[337, 337, w^3 - w^2 - 9*w - 5],
[353, 353, -w^3 + 2*w^2 + 5*w - 4],
[353, 353, w^3 - w^2 - 8*w - 3],
[353, 353, 3*w^3 - 10*w^2 - 7*w + 19],
[353, 353, -4*w^3 + 14*w^2 + 7*w - 22],
[361, 19, -3*w^3 + 11*w^2 + 6*w - 18],
[361, 19, 2*w^3 - 8*w^2 - 3*w + 17],
[367, 367, -4*w^3 + 12*w^2 + 13*w - 18],
[367, 367, 3*w^3 - 9*w^2 - 8*w + 13],
[383, 383, 5*w^3 - 17*w^2 - 12*w + 30],
[383, 383, 2*w^2 - 3*w - 5],
[383, 383, 3*w^3 - 9*w^2 - 11*w + 11],
[383, 383, -w^3 + 3*w^2 + w - 1],
[401, 401, 3*w^3 - 11*w^2 - 8*w + 23],
[401, 401, 4*w^3 - 14*w^2 - 11*w + 22],
[431, 431, 2*w^3 - 5*w^2 - 10*w + 8],
[431, 431, 6*w^3 - 20*w^2 - 15*w + 29],
[433, 433, -4*w^3 + 12*w^2 + 13*w - 17],
[433, 433, -3*w^3 + 9*w^2 + 8*w - 12],
[433, 433, 3*w^3 - 11*w^2 - 6*w + 20],
[433, 433, -2*w^3 + 8*w^2 + 3*w - 15],
[439, 439, -3*w^3 + 11*w^2 + 6*w - 19],
[439, 439, -2*w^3 + 8*w^2 + 3*w - 16],
[449, 449, w^3 - 4*w^2 + w + 11],
[449, 449, -w^3 + 2*w^2 + 7*w - 6],
[457, 457, w^2 - 4*w - 3],
[457, 457, 4*w^3 - 15*w^2 - 6*w + 27],
[457, 457, w^2 - 4*w - 2],
[457, 457, -2*w^3 + 6*w^2 + 8*w - 15],
[479, 479, -4*w^3 + 14*w^2 + 9*w - 29],
[479, 479, -w^3 + 5*w^2 - 6],
[487, 487, -w^3 + 6*w^2 - 3*w - 16],
[487, 487, -5*w^3 + 18*w^2 + 9*w - 29],
[503, 503, -2*w^3 + 7*w^2 + 8*w - 15],
[503, 503, 4*w^3 - 13*w^2 - 14*w + 20],
[521, 521, 3*w^3 - 10*w^2 - 9*w + 10],
[521, 521, -2*w^3 + 7*w^2 + 3*w - 16],
[529, 23, w^3 - 3*w^2 - 3*w - 1],
[599, 599, -4*w^3 + 12*w^2 + 14*w - 15],
[599, 599, 2*w^3 - 6*w^2 - 4*w + 5],
[599, 599, 4*w^3 - 13*w^2 - 12*w + 19],
[599, 599, w^2 - 6],
[601, 601, -4*w^3 + 12*w^2 + 11*w - 19],
[601, 601, 5*w^3 - 15*w^2 - 16*w + 24],
[607, 607, -w^3 + 4*w^2 + w - 13],
[607, 607, -w^3 + 4*w^2 + w - 2],
[617, 617, -4*w^3 + 13*w^2 + 11*w - 16],
[617, 617, 4*w^3 - 13*w^2 - 13*w + 20],
[625, 5, -5],
[631, 631, -3*w^3 + 8*w^2 + 11*w - 9],
[631, 631, 5*w^3 - 16*w^2 - 13*w + 24],
[631, 631, w^3 - 3*w^2 - 4*w - 1],
[631, 631, w - 6],
[641, 641, -3*w^3 + 12*w^2 + w - 17],
[641, 641, w^3 - 6*w^2 + 5*w + 18],
[673, 673, 4*w^3 - 15*w^2 - 6*w + 26],
[673, 673, -2*w^3 + 9*w^2 - 19],
[719, 719, 2*w^2 - 4*w - 5],
[719, 719, 4*w^3 - 14*w^2 - 8*w + 25],
[727, 727, 3*w^3 - 8*w^2 - 11*w + 8],
[727, 727, 6*w^3 - 18*w^2 - 19*w + 20],
[727, 727, -5*w^3 + 14*w^2 + 19*w - 17],
[727, 727, -5*w^3 + 16*w^2 + 13*w - 23],
[743, 743, 4*w^3 - 14*w^2 - 7*w + 23],
[743, 743, w^3 - w^2 - 8*w - 2],
[751, 751, 3*w^3 - 12*w^2 - 3*w + 23],
[751, 751, 3*w^3 - 12*w^2 - 3*w + 22],
[761, 761, 3*w^3 - 10*w^2 - 6*w + 22],
[761, 761, 3*w^3 - 10*w^2 - 6*w + 18],
[761, 761, -5*w^3 + 16*w^2 + 17*w - 25],
[761, 761, -2*w^3 + 5*w^2 + 9*w - 8],
[809, 809, -3*w^3 + 8*w^2 + 13*w - 3],
[809, 809, -2*w^3 + 6*w^2 + 3*w - 12],
[823, 823, -w^3 + 4*w^2 + 2*w - 2],
[823, 823, 2*w^3 - 7*w^2 - 5*w + 18],
[839, 839, 4*w^3 - 13*w^2 - 14*w + 27],
[839, 839, -7*w^3 + 23*w^2 + 19*w - 41],
[857, 857, -3*w^3 + 10*w^2 + 9*w - 12],
[857, 857, -w^3 + 4*w^2 + 3*w - 13],
[863, 863, 3*w^3 - 11*w^2 - 2*w + 20],
[863, 863, -2*w^3 + 4*w^2 + 13*w - 5],
[863, 863, 6*w^3 - 19*w^2 - 18*w + 27],
[863, 863, -2*w^3 + 5*w^2 + 6*w - 2],
[881, 881, -3*w^3 + 10*w^2 + 5*w - 17],
[881, 881, -3*w^3 + 8*w^2 + 13*w - 12],
[887, 887, -3*w^3 + 9*w^2 + 11*w - 9],
[887, 887, w^3 - 3*w^2 - w - 1],
[911, 911, 4*w^3 - 14*w^2 - 10*w + 17],
[911, 911, 5*w^3 - 17*w^2 - 12*w + 22],
[919, 919, -4*w^3 + 15*w^2 + 7*w - 30],
[919, 919, -2*w^2 + 7*w + 2],
[919, 919, w^3 - 5*w^2 + 4*w + 13],
[919, 919, 3*w^3 - 12*w^2 - 4*w + 20],
[929, 929, -4*w^3 + 13*w^2 + 9*w - 22],
[929, 929, 3*w^3 - 8*w^2 - 12*w + 12],
[961, 31, 4*w^3 - 12*w^2 - 12*w + 17],
[967, 967, -2*w^3 + 6*w^2 + 7*w - 2],
[967, 967, -3*w^3 + 8*w^2 + 15*w - 6],
[967, 967, w^3 - 2*w^2 - 5*w - 6],
[967, 967, 7*w^3 - 22*w^2 - 21*w + 36],
[983, 983, 5*w^3 - 17*w^2 - 13*w + 25],
[983, 983, w^3 - 5*w^2 - w + 15],
[991, 991, -6*w^3 + 21*w^2 + 13*w - 34],
[991, 991, 7*w^3 - 22*w^2 - 21*w + 33],
[1009, 1009, w^2 - 2*w + 3],
[1009, 1009, -2*w^3 + 7*w^2 + 4*w - 18],
[1031, 1031, -4*w^3 + 15*w^2 + 4*w - 26],
[1031, 1031, 3*w^2 - 8*w - 9],
[1033, 1033, 2*w^3 - 4*w^2 - 13*w + 3],
[1033, 1033, -5*w^3 + 15*w^2 + 19*w - 23],
[1033, 1033, w^3 - w^2 - 8*w - 10],
[1033, 1033, 4*w^3 - 14*w^2 - 7*w + 15],
[1049, 1049, -7*w^3 + 24*w^2 + 16*w - 34],
[1049, 1049, -5*w^3 + 16*w^2 + 18*w - 24],
[1063, 1063, w^3 - 4*w^2 + 2*w + 8],
[1063, 1063, w^3 - 2*w^2 - 4*w - 8],
[1063, 1063, 7*w^3 - 24*w^2 - 17*w + 39],
[1063, 1063, w^3 - 2*w^2 - 7*w - 6],
[1103, 1103, w^3 - 5*w^2 + 3*w + 17],
[1103, 1103, -w^3 + 5*w^2 - 3*w - 3],
[1103, 1103, -w^3 + 5*w^2 - 2*w - 6],
[1103, 1103, -2*w^3 + 8*w^2 + w - 19],
[1129, 1129, -6*w^3 + 18*w^2 + 20*w - 27],
[1129, 1129, -4*w^3 + 12*w^2 + 10*w - 17],
[1151, 1151, w^3 - w^2 - 7*w - 1],
[1151, 1151, 5*w^3 - 17*w^2 - 11*w + 29],
[1153, 1153, -2*w^3 + 8*w^2 + 3*w - 23],
[1153, 1153, -6*w^3 + 18*w^2 + 20*w - 25],
[1193, 1193, -2*w^3 + 5*w^2 + 9*w - 10],
[1193, 1193, 3*w^3 - 10*w^2 - 6*w + 20],
[1201, 1201, 6*w^3 - 19*w^2 - 16*w + 24],
[1201, 1201, 4*w^3 - 11*w^2 - 14*w + 9],
[1217, 1217, 7*w^3 - 22*w^2 - 21*w + 30],
[1217, 1217, 7*w^3 - 23*w^2 - 18*w + 39],
[1223, 1223, -2*w^3 + 9*w^2 - 15],
[1223, 1223, -3*w^3 + 9*w^2 + 13*w - 9],
[1223, 1223, -5*w^3 + 17*w^2 + 12*w - 34],
[1223, 1223, 4*w^3 - 15*w^2 - 6*w + 30],
[1231, 1231, -3*w^3 + 7*w^2 + 12*w - 3],
[1231, 1231, -8*w^3 + 26*w^2 + 21*w - 38],
[1289, 1289, -4*w^3 + 14*w^2 + 7*w - 24],
[1289, 1289, -w^3 + w^2 + 8*w + 1],
[1297, 1297, -4*w^3 + 12*w^2 + 11*w - 15],
[1297, 1297, -5*w^3 + 15*w^2 + 16*w - 20],
[1303, 1303, -3*w^3 + 10*w^2 + 7*w - 23],
[1303, 1303, -w^3 + 2*w^2 + 5*w - 8],
[1319, 1319, -8*w^3 + 28*w^2 + 15*w - 43],
[1319, 1319, w^3 + w^2 - 12*w - 12],
[1321, 1321, w^2 - 2*w - 10],
[1321, 1321, 2*w^3 - 7*w^2 - 4*w + 5],
[1367, 1367, -5*w^3 + 17*w^2 + 14*w - 26],
[1367, 1367, 2*w^3 - 8*w^2 - 5*w + 19],
[1399, 1399, -2*w^3 + 7*w^2 + 3*w - 4],
[1399, 1399, w^3 - 2*w^2 - 6*w - 6],
[1409, 1409, w^3 - 2*w^2 - 3*w - 3],
[1409, 1409, 3*w^3 - 10*w^2 - 5*w + 18],
[1409, 1409, -5*w^3 + 16*w^2 + 15*w - 22],
[1409, 1409, -3*w^3 + 8*w^2 + 13*w - 13],
[1423, 1423, 7*w^3 - 24*w^2 - 18*w + 44],
[1423, 1423, -8*w^3 + 26*w^2 + 21*w - 40],
[1433, 1433, -7*w^3 + 24*w^2 + 17*w - 45],
[1433, 1433, -3*w^3 + 12*w^2 + 3*w - 29],
[1439, 1439, -2*w^3 + 6*w^2 + 4*w - 3],
[1439, 1439, -4*w^3 + 12*w^2 + 14*w - 13],
[1471, 1471, -w - 6],
[1471, 1471, -w^3 + 3*w^2 + 4*w - 11],
[1471, 1471, w^3 - 2*w^2 - 8*w + 4],
[1471, 1471, w^2 - 5*w - 4],
[1489, 1489, -2*w^3 + 9*w^2 + 2*w - 17],
[1489, 1489, -6*w^3 + 21*w^2 + 14*w - 38],
[1543, 1543, -7*w^3 + 24*w^2 + 19*w - 45],
[1543, 1543, 2*w^3 - 3*w^2 - 15*w - 4],
[1543, 1543, -5*w^3 + 18*w^2 + 6*w - 26],
[1543, 1543, -3*w^3 + 12*w^2 + 7*w - 20],
[1553, 1553, -4*w^3 + 14*w^2 + 11*w - 20],
[1553, 1553, -3*w^3 + 11*w^2 + 8*w - 25],
[1559, 1559, 2*w^2 - 4*w - 3],
[1559, 1559, 4*w^3 - 14*w^2 - 8*w + 27],
[1567, 1567, -4*w^3 + 15*w^2 + 3*w - 20],
[1567, 1567, w^3 - 12*w - 10],
[1601, 1601, -3*w^3 + 10*w^2 + 10*w - 12],
[1601, 1601, 2*w^3 - 7*w^2 - 7*w + 18],
[1607, 1607, -7*w^3 + 23*w^2 + 20*w - 36],
[1607, 1607, -6*w^3 + 19*w^2 + 18*w - 26],
[1607, 1607, 2*w^2 - w - 9],
[1607, 1607, -2*w^3 + 5*w^2 + 6*w - 1],
[1609, 1609, -4*w^3 + 15*w^2 + 8*w - 24],
[1609, 1609, 4*w^3 - 15*w^2 - 8*w + 31],
[1663, 1663, -6*w^3 + 19*w^2 + 13*w - 26],
[1663, 1663, -7*w^3 + 20*w^2 + 26*w - 26],
[1681, 41, -5*w^3 + 15*w^2 + 15*w - 17],
[1697, 1697, -w^3 + 9*w + 6],
[1697, 1697, -7*w^3 + 24*w^2 + 15*w - 39],
[1721, 1721, -5*w^3 + 18*w^2 + 13*w - 38],
[1721, 1721, -w^3 + 6*w^2 - 5*w - 20],
[1721, 1721, -w^3 + w^2 + 6*w + 11],
[1721, 1721, 6*w^3 - 20*w^2 - 15*w + 24],
[1753, 1753, -3*w^3 + 9*w^2 + 13*w - 17],
[1753, 1753, w^3 - 3*w^2 - 7*w + 3],
[1753, 1753, -6*w^3 + 22*w^2 + 12*w - 45],
[1753, 1753, -6*w^3 + 21*w^2 + 16*w - 40],
[1759, 1759, -2*w^3 + 6*w^2 + w - 8],
[1759, 1759, 7*w^3 - 21*w^2 - 26*w + 33],
[1759, 1759, -3*w^3 + 10*w^2 + 10*w - 26],
[1759, 1759, -5*w^3 + 17*w^2 + 12*w - 37],
[1783, 1783, -5*w^3 + 18*w^2 + 11*w - 29],
[1783, 1783, 4*w^3 - 15*w^2 - 7*w + 26],
[1783, 1783, 3*w^3 - 12*w^2 - 5*w + 26],
[1783, 1783, -3*w^3 + 12*w^2 + 4*w - 24],
[1801, 1801, 8*w^3 - 24*w^2 - 27*w + 33],
[1801, 1801, 4*w^3 - 12*w^2 - 16*w + 19],
[1801, 1801, 4*w - 1],
[1801, 1801, 7*w^3 - 21*w^2 - 23*w + 33],
[1831, 1831, w^3 - 2*w^2 - 8*w + 2],
[1831, 1831, w^2 - 5*w - 2],
[1847, 1847, 6*w^3 - 20*w^2 - 15*w + 25],
[1847, 1847, w^3 - w^2 - 6*w - 10],
[1849, 43, -6*w^3 + 21*w^2 + 14*w - 37],
[1849, 43, -2*w^3 + 9*w^2 + 2*w - 18],
[1871, 1871, -4*w^3 + 13*w^2 + 8*w - 22],
[1871, 1871, 4*w^3 - 13*w^2 - 12*w + 15],
[1871, 1871, 4*w^3 - 11*w^2 - 16*w + 17],
[1871, 1871, w^2 - 10],
[1873, 1873, -w^3 + 5*w^2 - 20],
[1873, 1873, -4*w^3 + 14*w^2 + 9*w - 15],
[1913, 1913, -2*w^3 + 6*w^2 + 3*w - 4],
[1913, 1913, 5*w^3 - 15*w^2 - 18*w + 19],
[1913, 1913, 6*w^3 - 20*w^2 - 11*w + 30],
[1913, 1913, 9*w^3 - 30*w^2 - 23*w + 45],
[1993, 1993, -8*w^3 + 28*w^2 + 17*w - 43],
[1993, 1993, 4*w^3 - 13*w^2 - 16*w + 19],
[1993, 1993, -6*w^3 + 21*w^2 + 14*w - 36],
[1993, 1993, -2*w^3 + 9*w^2 + 2*w - 19],
[1999, 1999, -7*w^3 + 22*w^2 + 20*w - 30],
[1999, 1999, 4*w^3 - 11*w^2 - 13*w + 10]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^4 + 3*x^3 - 10*x^2 - 9*x - 1;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [3/8*e^3 + e^2 - 15/4*e - 9/8, e, 9/8*e^3 + 3*e^2 - 49/4*e - 51/8, -3/8*e^3 - e^2 + 15/4*e - 7/8, 1, -1/2*e^3 - e^2 + 6*e - 1/2, -1/4*e^3 - e^2 + 3/2*e + 7/4, -5/4*e^3 - 3*e^2 + 25/2*e + 7/4, -17/8*e^3 - 6*e^2 + 85/4*e + 83/8, -11/8*e^3 - 3*e^2 + 71/4*e + 49/8, 11/8*e^3 + 3*e^2 - 71/4*e - 17/8, 3/4*e^3 + e^2 - 19/2*e + 3/4, 9/8*e^3 + 4*e^2 - 37/4*e - 99/8, -9/4*e^3 - 6*e^2 + 45/2*e + 15/4, 5/8*e^3 + 2*e^2 - 17/4*e + 1/8, 2*e^3 + 5*e^2 - 22*e - 4, 3/2*e^3 + 3*e^2 - 20*e - 7/2, -3/2*e^3 - 3*e^2 + 20*e + 5/2, -17/8*e^3 - 5*e^2 + 105/4*e + 43/8, 7/4*e^3 + 4*e^2 - 45/2*e - 37/4, -1/2*e^3 + 9*e - 5/2, 1/2*e^3 - 9*e + 13/2, -15/8*e^3 - 5*e^2 + 87/4*e - 3/8, 3/2*e^3 + 4*e^2 - 18*e - 39/2, -29/8*e^3 - 9*e^2 + 157/4*e + 111/8, 4*e^3 + 10*e^2 - 48*e - 13, -2*e^3 - 6*e^2 + 17*e + 12, -13/4*e^3 - 8*e^2 + 81/2*e + 83/4, 11/2*e^3 + 13*e^2 - 64*e - 55/2, -1/4*e^3 + e^2 + 23/2*e - 53/4, 17/8*e^3 + 4*e^2 - 109/4*e - 3/8, -1/4*e^3 + e^2 + 17/2*e - 21/4, -6*e^3 - 16*e^2 + 62*e + 21, -15/4*e^3 - 10*e^2 + 71/2*e + 33/4, -25/8*e^3 - 10*e^2 + 121/4*e + 203/8, 1/8*e^3 + 2*e^2 - 1/4*e - 91/8, -17/8*e^3 - 6*e^2 + 93/4*e + 107/8, -5/4*e^3 - 2*e^2 + 41/2*e - 37/4, 17/4*e^3 + 10*e^2 - 101/2*e - 103/4, e^3 + 3*e^2 - 12*e - 8, 21/8*e^3 + 8*e^2 - 85/4*e - 111/8, 45/8*e^3 + 14*e^2 - 245/4*e - 159/8, 11/4*e^3 + 10*e^2 - 45/2*e - 97/4, 11/8*e^3 + e^2 - 75/4*e + 103/8, 1/2*e^3 + 2*e^2 - 9*e - 5/2, 1/2*e^3 - e^2 - 8*e + 25/2, -23/4*e^3 - 16*e^2 + 123/2*e + 161/4, 13/4*e^3 + 11*e^2 - 59/2*e - 115/4, -6*e^3 - 15*e^2 + 74*e + 31, 57/8*e^3 + 18*e^2 - 341/4*e - 259/8, 47/8*e^3 + 15*e^2 - 263/4*e - 261/8, -1/4*e^3 + 19/2*e - 21/4, -19/4*e^3 - 14*e^2 + 95/2*e + 117/4, -5/4*e^3 - 2*e^2 + 25/2*e - 21/4, -1/4*e^3 - 2*e^2 - 13/2*e + 43/4, -55/8*e^3 - 17*e^2 + 311/4*e + 269/8, 3/8*e^3 + 3*e^2 + 13/4*e - 161/8, 3*e^3 + 6*e^2 - 37*e - 13, -5/8*e^3 + 49/4*e - 73/8, 7/4*e^3 + 3*e^2 - 47/2*e - 17/4, 3/4*e^3 + e^2 - 25/2*e - 53/4, -9/4*e^3 - 5*e^2 + 55/2*e - 29/4, -7*e^3 - 17*e^2 + 79*e + 39, -5/4*e^3 - 5*e^2 + 7/2*e + 99/4, 29/8*e^3 + 12*e^2 - 149/4*e - 231/8, -29/8*e^3 - 12*e^2 + 149/4*e + 303/8, 39/8*e^3 + 11*e^2 - 231/4*e - 53/8, -27/8*e^3 - 9*e^2 + 179/4*e + 233/8, 2*e^2 + 9*e - 1, 9*e^3 + 24*e^2 - 101*e - 41, 29/4*e^3 + 21*e^2 - 149/2*e - 135/4, 5/8*e^3 - 17/4*e + 177/8, -7/4*e^3 - 3*e^2 + 43/2*e - 79/4, -13/8*e^3 - 6*e^2 + 49/4*e - 17/8, 6*e^3 + 18*e^2 - 57*e - 42, 33/8*e^3 + 9*e^2 - 177/4*e - 75/8, 29/8*e^3 + 12*e^2 - 137/4*e - 199/8, -1/4*e^3 - 3*e^2 + 1/2*e + 91/4, -27/8*e^3 - 9*e^2 + 115/4*e + 17/8, -9*e^3 - 24*e^2 + 95*e + 34, -71/8*e^3 - 23*e^2 + 391/4*e + 349/8, -1/2*e^3 - 2*e^2 - 4*e + 7/2, -21/8*e^3 - 7*e^2 + 105/4*e + 103/8, -21/8*e^3 - 7*e^2 + 105/4*e + 103/8, 17/8*e^3 + 8*e^2 - 61/4*e - 227/8, 11/4*e^3 + 5*e^2 - 67/2*e - 25/4, 11/4*e^3 + 7*e^2 - 51/2*e + 23/4, 47/8*e^3 + 16*e^2 - 243/4*e - 125/8, -17/2*e^3 - 22*e^2 + 100*e + 71/2, 53/8*e^3 + 17*e^2 - 325/4*e - 343/8, 37/8*e^3 + 11*e^2 - 245/4*e - 183/8, 13/2*e^3 + 18*e^2 - 69*e - 73/2, -35/4*e^3 - 22*e^2 + 205/2*e + 153/4, 1/4*e^3 + 3/2*e + 25/4, -e^3 - 3*e^2 + 11*e - 7, e^3 + 3*e^2 - 11*e - 22, -11/8*e^3 - e^2 + 83/4*e - 63/8, -1/2*e^3 - 4*e^2 - 2*e + 33/2, -23/8*e^3 - 6*e^2 + 151/4*e + 77/8, 23/8*e^3 + 6*e^2 - 151/4*e - 37/8, e^2 + e - 38, -3/2*e^3 - 5*e^2 + 14*e - 37/2, 3/4*e^3 + 2*e^2 - 15/2*e - 17/4, 7/8*e^3 + 4*e^2 - 11/4*e - 101/8, 13/4*e^3 + 7*e^2 - 77/2*e - 31/4, 19/8*e^3 + 4*e^2 - 103/4*e + 135/8, 25/4*e^3 + 19*e^2 - 121/2*e - 123/4, -7/4*e^3 - 5*e^2 + 29/2*e + 29/4, -17/4*e^3 - 11*e^2 + 91/2*e + 71/4, -37/8*e^3 - 16*e^2 + 161/4*e + 295/8, -7/4*e^3 - e^2 + 47/2*e - 79/4, 11/4*e^3 + 12*e^2 - 41/2*e - 157/4, -13/8*e^3 - 9*e^2 + 37/4*e + 295/8, 9/2*e^3 + 12*e^2 - 45*e + 15/2, 45/8*e^3 + 13*e^2 - 265/4*e - 271/8, -3/8*e^3 + e^2 + 55/4*e - 175/8, 87/8*e^3 + 28*e^2 - 483/4*e - 309/8, e^2 + 12*e + 12, 7*e^3 + 14*e^2 - 85*e - 19, 19/8*e^3 + 11*e^2 - 35/4*e - 353/8, -3/4*e^3 - e^2 + 21/2*e - 7/4, -e^2 - 3*e + 5, -7/2*e^3 - 7*e^2 + 44*e - 21/2, 1/2*e^3 - e^2 - 14*e - 15/2, 9/8*e^3 + 2*e^2 - 37/4*e + 141/8, 27/4*e^3 + 14*e^2 - 171/2*e - 61/4, -3*e^3 - 4*e^2 + 48*e - 4, 6*e^3 + 17*e^2 - 62*e - 21, 43/8*e^3 + 14*e^2 - 243/4*e - 33/8, -13/8*e^3 - 4*e^2 + 93/4*e + 255/8, -25/4*e^3 - 21*e^2 + 109/2*e + 175/4, -31/8*e^3 - 6*e^2 + 187/4*e - 139/8, -31/8*e^3 - 10*e^2 + 211/4*e + 245/8, 33/4*e^3 + 21*e^2 - 173/2*e - 111/4, 11*e^3 + 29*e^2 - 124*e - 50, 51/8*e^3 + 18*e^2 - 239/4*e - 225/8, 19/4*e^3 + 10*e^2 - 107/2*e - 49/4, 5*e^3 + 16*e^2 - 44*e - 43, -15/4*e^3 - 9*e^2 + 69/2*e + 17/4, -39/4*e^3 - 27*e^2 + 201/2*e + 197/4, -35/8*e^3 - 10*e^2 + 187/4*e + 1/8, -43/8*e^3 - 16*e^2 + 203/4*e + 193/8, 23/8*e^3 + 10*e^2 - 59/4*e - 357/8, 25/2*e^3 + 31*e^2 - 139*e - 147/2, 33/8*e^3 + 16*e^2 - 133/4*e - 395/8, -5*e^2 - 8*e + 29, -107/8*e^3 - 34*e^2 + 615/4*e + 513/8, 19/4*e^3 + 11*e^2 - 135/2*e - 81/4, 23/4*e^3 + 16*e^2 - 121/2*e - 29/4, 5/8*e^3 + e^2 - 13/4*e + 233/8, 17/8*e^3 + 8*e^2 - 53/4*e - 259/8, 5*e^3 + 11*e^2 - 58*e - 23, -27/2*e^3 - 37*e^2 + 141*e + 143/2, -33/8*e^3 - 10*e^2 + 141/4*e + 59/8, 17/8*e^3 + 8*e^2 - 49/4*e - 291/8, 3*e^3 + 3*e^2 - 36*e + 21, 75/8*e^3 + 30*e^2 - 351/4*e - 561/8, 49/8*e^3 + 14*e^2 - 281/4*e - 267/8, 13/2*e^3 + 21*e^2 - 61*e - 59/2, 11/8*e^3 - 71/4*e + 319/8, -39/8*e^3 - 13*e^2 + 195/4*e + 245/8, -5/2*e^3 - 4*e^2 + 35*e - 7/2, 9/4*e^3 + 7*e^2 - 35/2*e + 29/4, 21/4*e^3 + 13*e^2 - 115/2*e + 5/4, 7/4*e^3 + 2*e^2 - 55/2*e + 7/4, 133/8*e^3 + 44*e^2 - 721/4*e - 631/8, 7/4*e^3 + 5*e^2 - 7/2*e + 7/4, -5*e^3 - 11*e^2 + 64*e + 13, 37/8*e^3 + 10*e^2 - 241/4*e - 127/8, -5/2*e^3 - 7*e^2 + 28*e + 69/2, 7/4*e^3 + 5*e^2 - 41/2*e + 27/4, -1/4*e^3 + e^2 + 7/2*e - 157/4, -7/2*e^3 - 11*e^2 + 34*e - 5/2, -31/4*e^3 - 21*e^2 + 157/2*e + 125/4, -23/4*e^3 - 15*e^2 + 113/2*e + 65/4, -81/8*e^3 - 27*e^2 + 441/4*e + 387/8, -9*e - 9, -23/8*e^3 - 10*e^2 + 123/4*e + 101/8, 11/2*e^3 + 17*e^2 - 57*e - 121/2, 19/4*e^3 + 14*e^2 - 79/2*e - 185/4, 101/8*e^3 + 29*e^2 - 609/4*e - 383/8, -35/8*e^3 - 7*e^2 + 279/4*e - 23/8, 41/4*e^3 + 26*e^2 - 221/2*e - 251/4, 53/8*e^3 + 19*e^2 - 333/4*e - 615/8, -16*e^3 - 44*e^2 + 177*e + 66, 57/8*e^3 + 26*e^2 - 245/4*e - 475/8, -7*e^2 - 10*e + 58, -17/2*e^3 - 24*e^2 + 87*e + 89/2, -11/4*e^3 - 6*e^2 + 51/2*e - 11/4, 21/8*e^3 + 6*e^2 - 157/4*e - 111/8, -75/8*e^3 - 24*e^2 + 427/4*e + 345/8, 14*e^3 + 38*e^2 - 143*e - 54, 71/8*e^3 + 23*e^2 - 343/4*e - 141/8, -7/8*e^3 - 2*e^2 + 59/4*e - 259/8, 5*e^3 + 13*e^2 - 56*e - 62, -21/8*e^3 - 6*e^2 + 177/4*e + 223/8, 15*e^3 + 39*e^2 - 168*e - 61, 6*e^3 + 14*e^2 - 61*e + 5, 81/8*e^3 + 29*e^2 - 401/4*e - 323/8, -21/8*e^3 - 2*e^2 + 165/4*e + 63/8, 27/4*e^3 + 22*e^2 - 115/2*e - 229/4, 15/2*e^3 + 16*e^2 - 85*e - 35/2, 9/8*e^3 - 2*e^2 - 105/4*e + 333/8, -33/4*e^3 - 23*e^2 + 157/2*e + 211/4, -81/8*e^3 - 26*e^2 + 421/4*e + 419/8, 3/4*e^3 - 2*e^2 - 41/2*e + 91/4, -27/8*e^3 - 5*e^2 + 187/4*e + 17/8, -7/4*e^3 - 8*e^2 + 17/2*e + 161/4, -25/8*e^3 - 5*e^2 + 161/4*e + 91/8, -3*e^3 - 12*e^2 + 34*e + 53, 12*e^3 + 36*e^2 - 124*e - 76, -15/2*e^3 - 16*e^2 + 99*e + 47/2, 9*e^3 + 20*e^2 - 114*e - 26, -81/8*e^3 - 29*e^2 + 417/4*e + 435/8, -3/2*e^3 - 2*e^2 + 12*e - 33/2, -9*e^3 - 20*e^2 + 104*e + 29, -15/4*e^3 - 14*e^2 + 47/2*e + 173/4, 11/8*e^3 + 3*e^2 - 107/4*e - 233/8, -23/2*e^3 - 30*e^2 + 128*e + 73/2, 125/8*e^3 + 40*e^2 - 677/4*e - 583/8, -29/2*e^3 - 37*e^2 + 154*e + 99/2, 43/8*e^3 + 16*e^2 - 163/4*e - 265/8, -35/4*e^3 - 25*e^2 + 157/2*e + 141/4, 79/8*e^3 + 26*e^2 - 359/4*e - 205/8, 167/8*e^3 + 56*e^2 - 871/4*e - 733/8, 43/4*e^3 + 29*e^2 - 197/2*e - 137/4, 20*e^3 + 53*e^2 - 209*e - 83, -55/4*e^3 - 38*e^2 + 277/2*e + 357/4, -73/8*e^3 - 23*e^2 + 361/4*e + 387/8, -6*e^3 - 20*e^2 + 59*e + 43, 27/8*e^3 + 13*e^2 - 131/4*e - 433/8, 10*e^3 + 27*e^2 - 106*e - 40, 19/8*e^3 + 6*e^2 - 71/4*e + 55/8, 12*e^3 + 30*e^2 - 123*e - 51, 111/8*e^3 + 39*e^2 - 543/4*e - 669/8, -31/4*e^3 - 19*e^2 + 151/2*e + 13/4, 7/8*e^3 + 7*e^2 - 15/4*e - 141/8, -23/4*e^3 - 20*e^2 + 105/2*e + 285/4, -115/8*e^3 - 40*e^2 + 583/4*e + 473/8, -19/2*e^3 - 21*e^2 + 107*e + 55/2, -59/8*e^3 - 24*e^2 + 247/4*e + 505/8, -13/2*e^3 - 20*e^2 + 60*e + 73/2, -41/8*e^3 - 11*e^2 + 225/4*e - 5/8, -19/8*e^3 - 3*e^2 + 91/4*e - 135/8, -49/4*e^3 - 36*e^2 + 247/2*e + 303/4, 1/2*e^3 + 8*e^2 + 9*e - 125/2, -5/4*e^3 - 10*e^2 - 3/2*e + 83/4, 43/8*e^3 + 16*e^2 - 243/4*e - 377/8, -65/8*e^3 - 21*e^2 + 321/4*e + 155/8, -55/8*e^3 - 20*e^2 + 303/4*e + 325/8, -11*e^3 - 30*e^2 + 111*e + 43, 41/8*e^3 + 9*e^2 - 277/4*e + 13/8, -23/8*e^3 - 3*e^2 + 187/4*e - 35/8, -11/2*e^3 - 14*e^2 + 73*e + 87/2, 13*e^3 + 34*e^2 - 148*e - 54, -147/8*e^3 - 49*e^2 + 735/4*e + 481/8, 11/4*e^3 + 5*e^2 - 71/2*e - 97/4, -1/8*e^3 + 2*e^2 + 37/4*e - 269/8, -9*e^3 - 23*e^2 + 99*e + 27, -3/2*e^3 - 5*e^2 + 6*e - 9/2, 19/8*e^3 + 9*e^2 - 111/4*e - 633/8, -73/8*e^3 - 27*e^2 + 381/4*e + 123/8, -47/4*e^3 - 34*e^2 + 239/2*e + 321/4, -5/2*e^3 - 4*e^2 + 23*e - 3/2, -23/8*e^3 - e^2 + 163/4*e - 227/8, -55/8*e^3 - 25*e^2 + 227/4*e + 541/8, -95/8*e^3 - 35*e^2 + 415/4*e + 621/8, -20*e^3 - 50*e^2 + 215*e + 87, 15/4*e^3 + 11*e^2 - 57/2*e - 177/4, 45/4*e^3 + 29*e^2 - 243/2*e - 303/4, 87/8*e^3 + 31*e^2 - 419/4*e - 717/8, -9/2*e^3 - 14*e^2 + 61*e + 133/2, 81/8*e^3 + 25*e^2 - 421/4*e - 507/8, 75/4*e^3 + 52*e^2 - 407/2*e - 349/4, -4*e^3 - 14*e^2 + 34*e - 2, -103/8*e^3 - 34*e^2 + 587/4*e + 381/8, 13/2*e^3 + 17*e^2 - 83*e - 117/2, -2*e^3 - 2*e^2 + 26*e - 50, 27/2*e^3 + 34*e^2 - 156*e - 89/2, -39/8*e^3 - 11*e^2 + 279/4*e + 301/8, 9/4*e^3 + 6*e^2 - 35/2*e - 95/4, 63/8*e^3 + 21*e^2 - 335/4*e - 445/8, -21/4*e^3 - 9*e^2 + 145/2*e - 25/4, 33/8*e^3 + 6*e^2 - 245/4*e - 35/8, -9/4*e^3 + 81/2*e - 17/4, 49/4*e^3 + 32*e^2 - 257/2*e - 247/4, 9/4*e^3 - 81/2*e + 145/4, 29/4*e^3 + 20*e^2 - 133/2*e - 163/4, -39/8*e^3 - 13*e^2 + 199/4*e + 101/8, -15/4*e^3 - 10*e^2 + 73/2*e + 25/4, 69/4*e^3 + 49*e^2 - 349/2*e - 363/4, 57/8*e^3 + 16*e^2 - 277/4*e - 3/8, -77/8*e^3 - 23*e^2 + 497/4*e + 399/8, 119/8*e^3 + 37*e^2 - 707/4*e - 477/8, 11*e^3 + 23*e^2 - 138*e - 59, -31/8*e^3 - 4*e^2 + 267/4*e - 355/8, -15/8*e^3 - 3*e^2 + 107/4*e + 229/8, 15/8*e^3 + 3*e^2 - 107/4*e + 235/8, 5/8*e^3 + 8*e^2 + 43/4*e - 647/8, 25/8*e^3 + 2*e^2 - 193/4*e - 203/8];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {17, 17, 2*w^3 - 6*w^2 - 7*w + 8}>] := -1;

// EXAMPLE:
// pp := Factorization(2*ZF)[1][1];
// heckeEigenvalues[pp];

print "To reconstruct the Hilbert newform f, type
  f, iso := Explode(make_newform());";

function make_newform();
 M := HilbertCuspForms(F, NN);
 S := NewSubspace(M);
 // SetVerbose("ModFrmHil", 1);
 NFD := NewformDecomposition(S);
 newforms := [* Eigenform(U) : U in NFD *];

 if #newforms eq 0 then;
  print "No Hilbert newforms at this level";
  return 0;
 end if;

 print "Testing ", #newforms, " possible newforms";
 newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *];
 print #newforms, " newforms have the correct Hecke field";

 if #newforms eq 0 then;
  print "No Hilbert newform found with the correct Hecke field";
  return 0;
 end if;

 autos := Automorphisms(K);
 xnewforms := [* *];
 for f in newforms do;
  if K eq RationalField() then;
   Append(~xnewforms, [* f, autos[1] *]);
  else;
   flag, iso := IsIsomorphic(K,BaseField(f));
   for a in autos do;
    Append(~xnewforms, [* f, a*iso *]);
   end for;
  end if;
 end for;
 newforms := xnewforms;

 for P in primes do;
  xnewforms := [* *];
  for f_iso in newforms do;
   f, iso := Explode(f_iso);
   if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then;
    Append(~xnewforms, f_iso);
   end if;
  end for;
  newforms := xnewforms;
  if #newforms eq 0 then;
   print "No Hilbert newform found which matches the Hecke eigenvalues";
   return 0;
  else if #newforms eq 1 then;
   print "success: unique match";
   return newforms[1];
  end if;
  end if;
 end for;
 print #newforms, "Hilbert newforms found which match the Hecke eigenvalues";
 return newforms[1];

end function;